Load carrying potentiometer



April 15, 1958 D. KATZ LOAD CARRYING POTENTIOMETER Filed June 27,- 1955 4 Sheets=Sheet 1 INVENTOR April 15,1958 D. KATZ 2,831,158

LOAD CARRYING POTENTIOMETER Filed June 27, 1955 4 Sheets-Sheet 2 1 N VENTOR April 15, 1958 D. KATZ 2,831,158

LOAD CARRYING POTENTIOMETER Filed June 27, 1955 4 Sheets-Sheet 3 INVENTOR April 15, 1958 Filed June 27, 1955 WIIIIIIIIIYIIIIIAS ES D. KATZ LOAD CARRYING POTENTIOMETER 4 Sheets-Sheet 4 INVENTOR United States Patent LOAD CARRYING POTENTIOMETER David Katz, Wilmington, Del.

Application June 27, 1955, Serial No. 518,240

9 Claims. (Cl. 323-74) This invention relates to electrical devices of the potentiometer or voltage divider"class. It is an object of this invention to provide novel devices of this type which can be used to measure off a specified fraction of a voltage or of a resistance, even while the device carries a load, without detriment to the accuracy of the measurement by virtue of said load or of variations therein. Additional objects and achievements will become apparent as the description proceeds.

The potentiometer or voltage divider is a Well-known device in electrical measurements or instruments. The elementary form thereof, as well as its principle of operation is described in many textbooks or manuals. See, for instance, Analog Methods in Computation and Simulation by Walter W. Soroka (McGraw-Hill, 1954), page 48, or Radio Communication Pamphlet No. 40 (Government Printing Ofiice, 1922), pages 55 to 57 and 84 to 86.

Devices of this type have numerous applications in the design of electrical instruments, in electrical measurements, in radio communication, etc. A particular and relatively recent field of application is in the building of calculating machines and servo-mechanisms. See, for instance, Servomechanism Practice by William R. Ahrendt (McGraw-Hill Book Co., 1954), chapter 2, or the Soroka book above referred to.

A serious limitation, however, on the use of potentiometers for the last mentioned purpose is their inherent property of giving an accurate voltage division only if the principal resistance of the device carries no side load-s. If the device be used for measuring an unknown E. M. F., the latter must be shunted across certain points of the graduated potentiometer resistance, but the points selected must be such that no current whatever flows through the shunt. A galvanometer included in the shunt must read zero. If this condition is violated, the reading is no longer accurate.

Again, for many purposes, devices of the above type are of interest for dividing a given voltage or a given resistance in specified ratio, rather than for measuring an unknown voltage. For instance, a circuit may be given wherein the drop of potential across a certain, principal resistance is 100 volts, and it may be required to draw off from it an E. M. F. of 40 volts. If the principal resistance is uniform and graduated, it is easy to locate two points on it whose difference of potential is precisely 40 volts. But the instant that a load (another instrument, resistance, lamp, etc.) is attached to these two points, to make use of the 40 volts, the value of the latter changes (usually drops), and we are no longer certain what the voltage is under which the load is operating.

Attempts to overcome this difficulty have already been made in the art. Chapter 2 of the above cited textbook of Soroka describes several of these. They have, however, certain other drawbacks. The solution wherein vacuum tubes are interposed between the voltage divider and its load has the disadvantage of introducing vacuum tube elements into the circuit, with all their attendant complications, difiiculties and costs. For instance, a vacuum tube requires being heated, and its accuracy for use in analog computers depends on the straightness of the I E curve. Other devices hitherto proposed have a limited range of utility or are again too complicated for economical use.

Now, my invention provides a novel voltage-divider or potentiometer device, whose deviation from standard structure involves only resistances and switches, and which will give accurate voltage-division under load, regardless of the magnitude of such load or of variations in such magnitude during use.

For a better understanding of my invention, reference is now made to the accompanying drawings, in which:

Fig. 1 is an electrical diagram which aids in explaining the principle of my invention.

Fig. Z-is an electrical diagram showing a practical em- I bodiment of the principles taught by Fig. 1, using a special form of resistor B which is shown in greater detail in Fig. 3. Fig. 2 further illustrates the use of this embodiment in a cascaded system, in certain special applications, of this invention.

Fig. 3 is a top view of a laminated resistor B which constitutes an integral part of this invention.

Fig. 4 is a section through Fig. 3 at plane 4, 4, looking downwards.

Fig. 5 is an electrical diagram showing the manner of coupling a voltmeter to resistor 7 of Fig. 2.

Fig. 6 is an electrical diagram showing how the voltage across resistor 7 may be measured by means of a potentiometer circuit.

Fig. 7 is a front view of a modified form of this invention, wherein the principal resistors A and B are of a curved or cylindrical shape.

Fig. 8 is a diagrammatic representation of a modified geometrical disposition of the essential resistors in the layout of Fig. 2.

Fig. 9 is a modified form of resistor A employed in the layout of Fig. 2.

Fig. 10 is a front view of a modified form of this invention, employing as resistor A the modified form of Fig. 9.

Fig. 11 is a top view of the layout shown in Fig. 10.

Fig. 12 shows a detail of propelling nut 36 employed in Fig. 10.

Fig. 13 is a plan view of a modified form of the laminated resistor B shown in Fig. 3.

Figs. 14 and 15 show a modified form of resistor B, Fig. 14 being a plan view and Fig. 15 a section through line 15-15 of Fig. 14.

Fig. 16 is a plan view of an assembly including resistors A and B in a modified form.

Fig. 17 is a side view of the assembly shown in Fig. 16.

Fig. 18 is a detail, in perspective, of the sliding member 13 used in Fig. 17.

Basic theory Referring now to Fig. 1 in detail, three types of lines are to be distinguished there. The heavy lines represent electrical resistances, fixed or variable. Lighter lines represent electrical connecting wires of negligible resist ance, or whose resistance does not enter into the calculations. These lighter lines are permanently attached to the other resistances at certain points (indicated by heavy dots) or they may be plugged into the other resistances (indicated by heavy arrows). Discontinuous, light dimension lines (terminating in light arrows) indicate the extent of the various electrical quantities indicated, such as the extent of resistances A, B and R, or of potential difierences E and e. battery makes a closed circuit with the variable resistance Patented Apr. 15, 1958 A conventionally indicated cell or e system AB. E is the terminal voltage of this-cell or battery.

Restating now my invention in different words, the principal aimof the device is to enable us to la-y off on line AB a voltage 2 which is less =than E but w'hose ratio to -E has been assigned to us. In other-words, given'a fractional number x, x 1 );"the problem is-to produce an'ele'ctric'al instrument on which we canreadily lay off a voltage ezxE, wherein E is the terminal voltageof the system, the actual value of E itself being-immaterial, and perhaps unknown. g 7

It will-be noted that'inFig. -1 wehavea resistance-system comprising a variable resistance A in series with a parallel systemconsisting of fixed resistance R and varia'ble resistance B. Hereinafter We shall call the -series AB, the basic resistance line of the system; resistance R, the shunt'or load; and the-effective resistance of the entire system AB'R, the total resistance under E.

Two basic conditions will now .be imposedu'pon the device of my invention.

(1) The total resistance under E must have the value R and must remain constant, no matter whatthe position of the shunt R .is along the basic line AB.

(2) When the shunt is replaced by a load, theresistance of the latter must likewise have the value R and must be kept constant no matter how conditions within the load circuit are modified.

It'fo'llows at once from condition '2, that shunt R may be replaced by another resistance :systern ARR, which obeys condition 1. Then the shunt R inthe latter, may be replaced by a third similar :system. Andso on. In this fashion .a cascade of devices .of this nature can be assembled, while yet maintaining the total'resistance-under E .at the value R. This value .R, we :shall hereinafter designate as the basic resistance value or .the basic unit.

Now, to develop the problennlet U=thextota1 resistance of system AB R, and let R the resistance of Ethe parallel system BR.

It follows that, given Now, we have times. obtain and finally, AB=xR It'will be noted that as x O, A R, H 0 and e O as x l, A 0, B and e E, and when x=0.5, A=0.5 R, B=R and 'e==0;5 E

When these figures *areapplied to the diagram of Fig. '1. it will be clear that as x=0, the two plugging-in arrows of shunt R come together on line AB at the point thereon marked R. As x Lthe left-hand arrow recedes to the extreme left of the A-B line, whereas the arrow on the right, while still in contact with the corresponding arrow from the battery, runs off the diagram on the righthand side toward infinity.

Observing from Fig. 1, that the shunt R is plugged in on basic line A-B at points a and'b, the practical problem on hand reduces to the following:

Given a positive fraction x and a basic resistance unit R, to devise a basic line of resistances A and B so that by a simple mechanical motion (translation or rotation) proportional to x, we may be able to locate on the basic line points a and b so that 1 p mR A-R(.1 .m) and B-.

For a-practical- 'solution to this problem according to this invention,

we now turn to Fig. '2.

Practical embodiment V tional to the displacement of the movable contact point.

- In the particular modification-chosen for illustration in Fig. 2,'the-axis ofrheostat 11 is -a straightline.

Spaced from rheostat 1, and with its contact-making edgesinta-fixed disposition with respect to rheostat 1, is a .spe'cial curve resistance 5, to be described in further detailzhereinbelow. In the particular modification chosen for illustration in Fig. -2, the said contact-making edge of 5 is 'astraight line, parallel -to the axis of 1 and directly underneath it.

:Between resistances 1 and 5 and in-electrical contact -with each of them is a sliding member 3, made of conducting material and built *sulnciently massive to have negligible resistance. In the particular modification illustratedin Fig. 2, this member has the shape of a tuning fork, "but it may in practice be modified to have any suitable shape so long'as it is adapted to perform the folowing tfour functions:

1) It must be adaptedto move in a straight line or circle parallel 'to the linear or circular axis, respectively, of :rheostat 1.

-(2) Itmustlbe adaptedto sh0W 0n -a scale the amount of its displacement from position zero.

(31) Itrnustmake :apoint contact with rheostat '1 in any or its positions, th'e location of which point, however, will vary with the position of the sliding member 3.

(4) vIt must make -a line contact with the aforementionedlcontact making edge of the special-curve resistance-5.

In Fig. 2, function 3 is achieved by a spur or similar contact :point 32 fixed on the upper leg 31 of sliding member3.

Function 2 is achieved by a stationary scale 2, cooperating with the forward end of one of the legs of memher 3 or with a special pointer (not shown) fastened to one of said legs in a convenient position. Scale 2 represents the x-axis. In Fig. 2, the zero point of this scale is on the left and its range from 0 to 1.0 coincides in length with the length of rheostat 1 (as measured from its zero point to the point where its resistance is R). If rheostat'l is uniformly wound, scale 2 must likewise be uniformly graduated; that is its reading must be a linear fundtionof the-displacement of member 3. Other styles of subdividing "scale 2 are possible (for instance,

logarithmic), providing the design of resistances 1 and is correspondingly modified.

Function 4 is achieved by the lower leg 33 of sliding member 3, which is in continuous line contact with the upper edge 52 of special-curve resistance 5.

The words point contact and line contact above are used in a relative sense, and do not take into account the third dimension of resistances 2 and 5, that is, the dimension perpendicular to the plane of the drawing. In practice, this third dimension does come into action, and the point contact with rheostat 1 is actually a line contact or an oblong rectangle or ellipse contact, with the longer dimension perpendicular to the plane of the drawing. Likewise, the line contact with special-curve resistance 5 is actually a rectangular area contact, the area being in a plane perpendicular to the plane of the drawing.

The laminated resistance Referring now to Fig. 3, the special-curve resistance 5 has been specially invented by me and designed to serve the purposes of this invention. In its essence, it is a pile of laminated, conductors of varying heights and assembled to pave out with their butt ends a smooth plane surface of uniform width in the dimension perpendicular to the plane of the drawing, while the opposite ends define a specified mathematical curve. The individual laminae or plates 51 are actually much finer than shown in Fig. 3, the latter being merely symbolic or diagrammatic in this respect. They may consist of mutually insulated metal foil of rectangular shape, or of plastic foil coated uniformly or imprinted with a conducting film (as for instance, a metallic or graphite film). Or they may consist of fine wires suitably surfaced with a film of lacquer or with an insulating oxide film. In either event, the laminae are assembled to form a plate of uniform thickness and of a width corresponding to the length of scale 2 and are held together in this position by plastic plates or paper binding sheets 56 (see Fig. 4). The butt ends of the laminae are arranged to form a rectangular, relatively plane area 52 on one edge of the plate. The plate is then held tight in a jig, and edge 52 is machined, ground and polished to produce a smooth and true plane, rectangular surface. The thickness of the plate (i. e. the dimension perpendicular to the plane of the paper) may be varied from one design to another, but will generally be small compared to the length of edge 52, which in turn is equal to the length of the range of 0 to 1.0 on scale 2. In fact, in some designs, as explained in connection with Fig. 7 hereinbelow, the laminated plate is made thin enough to be capable of being bent into a semicircular cylinder whose base is defined by edge 52.

The opposite ends of the laminae are then cut away, machined and ground or polished to produce a curving edge 53 whose shape corresponds to a designated mathematical curve based on edge 52 as x-axis. In the preferred form wherein scale 2 is divided into uniform, linear divisions, the curved edge 53 follows a parabola. More particularly, the curve must be defined by the equation y=Kx where K is a constant to be determined hereinbelow.

The thus formed and polished edge 53 is then made conductively integral (as by soldering) with a conducting strip 54, which is sufficiently massive to have negligible resistance. The right-hand end of this strip (assuming that the origin of x is at the left) is then extended forward or downward, to provide a contact zone 55 for connecting to electric wiring or other electrical apparatus.

The performance of this type of resistance can now be readily explained. If a fiat-surfaced conducting member 33 is brought into uniform and intimate electrical contact with the butts of the laminae 51 along a portion of edge 52, and if a potential difference V is applied between element 33 and end 55, current will flow (as shown by the arrows) across edge 52, through the individual laminae contacted by it, to conducting strip 54 and out at 55. The conductance (reciprocal of resistance) encountered by this current will be determined by the summed conductance of the laminae contacted by element 33. The more of these laminae are contacted, the less the total resistance. On the other hand, the longer each lamina, the greater is its individual resistance.

In the particular design of Fig. 3, if the range of contact covers the space between point x and point 1.0 on the scale, the integrated resistance of the laminae in action equals This is the required mathematical expression for resistance B in Fig. l.

The device as a whole Let us now return to Fig. 2.

A battery 4 is connected across the system by contacts at point 0 of the rheostat 1i and point 55 of resistance 5 If key 41 in the same circuit were closed, current would flow as indicated by the arrows, from the battery through point 0, along rheostat 1, through contact point 32 and leg 31 of member 3, then further across leg 33 into certain laminae of resistance 5, then out along member 54 to point 55 and back to the battery. If the position of the spur 3,2 (and corresponding edge of leg 33) is at x on the scale, the portion of resistance 1 thrown into the circuit is A=R(lx), While that of 5 is Ra: B l -a:

The circuit then consists simply of a battery and two esistances A and B in series.

However, the circuit contains further a double-poledouble-throw switch 6, and a fixed resistance 7 whose value is again equal to our basic unit R. The latter is connected across one pair 61 of the outer terminals of switch 6, while the middle pair 60 of the latter are connected respectively to point 34 of member 3 and point 55 of member 5.

Before closing key 41, switch 6 is thrown to the right,

' to close the same upon said pair of terminals 61. Thus resistance 7, of fixed value R, is shunted across variable resistance B, which satisfies the basic diagram of Fig. 1.

Key 41 may now be closed, and sliding member 3 may be moved to any desired position x (within range 0 to 1.0 on the scale). And no matter what this position be, the total resistance of the circuit is R (constant); the value of A is R(lx); the value of Ra: B is m the terminal E. M. F. of the battery is E (constant, except for temperature variations and degradation of the battery itself), and the drop of potential between points 34 and 55 is e=xE.

The principal object of this invention has thus been achieved. We have a device by which we can readily lay off a fraction xE of a specified voltage E, by merely moving a sliding member 3 to position x along a graduated scale.

If it is desired to vary E itself, this can be achieved by including a variable resistance in the battery circuit in series with rheostat 1, but externally to the system included between the points 0 and 55, or we may shunt a variable resistance across the terminals of the battery itself. These variations being readily understood by those skilled in the art, they are not shown on the drawings. The value of the basic unit R itself cannot be tampered with, once the device is built; but it may be selected at any convenient value while the design is still on the draft board.

,. gas -ms The device up to this-point .may be looked upon .as a voltage divider, which receives from a battery a ,given terminal volt age 1E, and delivers at its taps .34 and 55 any desired fractional value of said voltage from .to E. But .accurate performance of the device can be obtained only if resistance 7 is maintained across the outer terminals of DPDT switch .6. If the delivered voltage e=xE is to be put to a practical use, it will obviously be necesary to transferit to the desired .load (grid of a vacuum tube,,meter servomechanism, etc.). Thiscan be achieved simply by connecting the load and switch 6 to the other pair of outer terminals, 62, provided care is taken to ballast the load with variableresistances until its total resistance equals exactly R. How this is to be done in any particular case will, obviously, depend on the nature of the load and the design of its circuit. But for the purpose of illustration, Fig. 2 contains a showing of how this can be done *in 'a particular case.

Cascaded voltage divider The particular case or" loading illustrated in Fig. 2 deals with cascaded potentiometers or voltage dividers. Suchcascades are useful'in the design of servomechanisms or analog computers. See for instance Soroka, above cited, atpages 51 to 52 (Figs. 2.9, 2.10 and 2.11) and at page 128 (Fig-4.1). In the devices there discussed, loading effects are prevented by resorting to electron tube devices (cathode-followers) or are ignored, which means that up to a certain degree the error introduced by loading is tolerated, while above that degree the device is simply not used.

Now according to my invention as shown in Fig. 2, terminals 62 of switch 6 are connected directly across the points 0 and 55 of another similar potentiometer device II. Here again points'34 and 55 are connected to the middle terminals 600i a DPDT switch 6, whose outer terminals 61 must be spanned by a resistance of value.R. But instead of using a new resistance unit for this purpose, terminals 61 may be simply connected to resistance 7 by tapping in on lead wires 63.

If desired, a third similar potentiometer unit may likewise be switched in at terminals 62 of the second switch 6, and likewise a fourth and a fifth unit, provided it is remembered always to have the last switch 6 closed upon resistance 7, or upon an equivalent resistance who-se'value is R.

The positions of the slides 3 in the various successive units need not be identical. And if x x x represents the positions of these slides in the successive units, the voltage delivered at the taps of the final unit will be e =x E, e =x x E, e =x x x E, etc., depending on how many units have been put into the cascade.

Cascades of this type can be used in servomechanisrns to achieve control of a machine or of a chemical or manufacturing process, wherever it is desired to regulate a machine or process-according to the mathematical'prodnet of 2 or more variable factors within the machine or process. For this purpose, each factor may be con nected -individually'( by means of mechanical or electrical mechanism) to .a sliding member 3 of one of the potentiometer unitsin'the cascade, whereby to move said sliding member along the x-axis in proportion to the value of the factor. The terminals of the final unit in the cascade then deliver avoltage which is proportional to .the product of the individual factors. These terminals 61 of the final switch 6 in the cascade may then be connected to "a suitable electrical mechanism for feeding back the proper control or regulation into the ma chine or process, as the particular case on hand may require.

The last 'mention'ed electrical feed-back mechanism may ;be made to replace resistance 7 .at the terminals 61, provided it is'designed to have a 'total resistance equal to R and to maintain this value constant throughoutits operation. But a simpler procedure is to retain resistance 7 and connect its terminals to the ,grid ,of an electronic vacuum ,tube. The plate of the latter may ,then be connected, through a local battery, to any desired mechanism for performing any function whatever. For this purpose, resistance 7 is shunted by a switch I71, and the manner of its use is illustrated in Figs. ,5 and -6.

Cascades of the above type may also be used inca'lculating machines. For instance, if we are given ,three fractions x x and x and we have a cascade of three potentiometer units, we may move manually the sliding member? of ,each of these to lay ofi on its respective scale one of these fractions. The product, as already stated, is then represented by the voltage drop e across resistance ,7 in the final unit of the cascade. The problem now is how to .read this value e.

As shown in Fig. 5, this may be achieved simply by connecting a voltmeter across the outer terminals of switch 7-1. Assuming that the resistance of the voltmeter is highzcompared to R, its reading will represent with reasonable :accuracy the value of .2. .Then x x x =e/E. Furthermore, if ,E is known .in advance, the voltmeter maybe calibrated to read offthe product 1: 3 :6

In Fig. 6, the same result is achieved, with higher accuracy by the aid of a no-current, measuring circuit. Forthis purpose, an ordinary potentiometer including battery81, principle resistance 82 and movable contactpoint or plug '83 is employed. The potentiometer'mayfb'e of the sliding wire type or of the circular coil type, but in eitherevent the device is provided with a scale to indicate the degree of movement or the fraction of terminal voltage chopped off bythe plug 83. A galvanometer G, and lead wires connecting the device to switch 71, complete the circuit. But to protect thega'lvanometer against unexpected, excessive currents, a voltmeter V may be inserted parallel to G. A key 84 held back bya spring, or a push 'button device '85, then serves to keep V included in and .G excluded fromthe circuit most of the time. In making a measurement, plug 83 is moved experimentally along resistance 32, until a point is found wherein essentiallyno current flows throug'h'V. The key is "then pressed to exclude V and to insert G in its place, and finer adjustments in the position of contactor '83 are made, until no current flows through the galvanometer.

Systems for performing other mathematical operations such as division, squaring, cubing, taking roots, solving polynomial equations, etc., may be designed in similar manner by the aid of my novel voltage dividing device, and servo-mechanisms for carr ing out analogous functions automatically may likewise be constructed. It will thus be clear that my novel potentiometer device has numerous potential applications in practice'the details of which need not be gone into any further .at this point. But now, to complete the disclosure, we shall go 'back'to the special-curveresistance 5 and discuss its theory ,and principles of design.

Theory ofthe laminated resistance Going back to Fig. 3, "let "X- represent theabscissa of any point on curve 53 in cm., and let y represent its ordinate. Let L represent the length of edge 52, from x=0 to x=1. Then x=X/L. As x varies from O to l, X varies from-O to L. The laminae-contacted by leg 33 of slide 3 when its end is at X are those included in the region between X and L. The conductance C ofthis group of laminae is the sum of the conductances of the individual laminae. The resistance B of the group is the inverse of the conductance; "B: 1/6.

Let AA=the cross sectional-area in square cm. of one lamina, A'X -it-s thickness in cm. along the X-axis, b=its width'in cm. perpendicular to the plane of the drawing. and at its height. Let c=thespecific conductance ofthc conducting materialof the lamina, and jits .area ratio, that is, ithe-ratioof conducting area to total areain a given section. (For instance, if our laminae are made 9 up of plastic foil coated with a metal film, and if the thickness of the metal film is A of that of the plastic film, the cross sectional area of each lamina is bAX square cm. of plastic material and bAX square cm. of metal; hence j= /s.) As for 0, its value is l/p, wherein p is the specific resistance of the metal employed, expressed in ohms per cm. length for each square cm. of cross section. v

The values of p are recorded in the literature or may be determined in the laboratory for any particular novel conducting medium. See for instance, Handbook of Chemistry and Physics, nineteenth edition, page 1337 to 1343. By way of illustration, a few such -values are noted here.

Silver l.5-l.7 10 Copper 1.54-1.68Xl' Aluminum 2.6-3.0 10- Magnesium 4.l-5.0 l0 Manganin alloy 4()--70 10- Carbon (in lamp filament) 4000 l0 The conductance, then of 1 lamina is ojAA jcbAX The conductance of the pack between abscissas X and L is approximately C jcbdX The broader case wherein y is given different forms in terms of X to obtain corresponding varying Us is discussed further down in this specification. For the purpose of the potentiometer problem dealt with in the instant case, it is sufficient to set y=kX Then the definite integral is Comparison of (4) and 5) shows that our problem I will be satisfied if Replacing 1 F y P the formula becomes -12 1 R- jb Formula 7 can be used as the basic law for designing resistor B for the purpose of this invention. For instance, let us assume that we desire to make the length of scale 2 about 10 inches, and the thickness of plate 5, 1 cm; then we select L=25 cm. and b=l. Also, since y=kX the highest ordinate in the plate is at X=L. For practical' purposes, it will be desirable to keep the height of 10 this terminal ordinate y at about the same value as length L; therefore we choose then y =L. Equation 7 then becomes simplified into If we try to make our laminae out of alternating sheets of metal foil and plastic, we shall find that the values of R obtainable are too small for practical purposes. For instance, if we make j= and p=2.8 10' (using aluminum foil), then R becomes 14 microhms. With such a low value of R, the resistances of the lead Wires and contact points could not very well be neglected.

So instead of alternating plastic and metal foil, we shall use a plastic film (say, viscose or cellulose acetate film) spray-coated with a suspension of carbon black in a suitable liquid (which is then evaporated). The coating may cover the entire face of the film; yet, its extreme thinness compared to that of the film may reduce j to a very small value, say j= ,4 Assuming now =4OGO l0' we obtain R=0.l60 ohm.

This is a much more reasonable value of R for practical use, but We can increase it still further by reducing j and increasing p by various means. For instance, instead of spray-coating the cellulosic film over its entire surface, we may simply imprint upon it straight lines, 1 per cm., of say 0.5 mm. Width. We then cut the film into strips 1 cm. wide, obtaining now a j-value 20 times as small as before. Then R becomes 3.20 ohms.

We may also dilute the concentration of the carbon black in the printing paste, thereby increasing p to very high values. In that fashion very large values of R can be obtained.

Still higher values of R may be obtained by using; mutually insulated laminae made up of semi-conducting;

plastics. For instance, by incorporating carbon black into. the melts or solutions from which plastic films (viscose, cellulose acetate, methyl methacrylate, etc.) are formed, compositions may be produced having -values of the order of 10* to 16 ohms or even higher. Assuming for instance a material having =5 X10 and assuming laminae 1 cm. wide and having insulating coatings of negli gible thickness, We may set i=1 and 17:1; then (if kL=1) R=50,000 ohms. By altering the values of j, b and kL, further variations in the value of R may be achieved.

It will be clear that the value of R can be varied within very wide limits, to suit practically any particular problem on hand.

Equivalents and modifications It will be noted that if L and y are equal, then lcL=l and the actual values of L and y become immaterial insofar as the value of R is concerned. L,h0wever, is material in determining the fineness with which rheostat l and resistor 5 may be adjusted and read. The longer 1., the more laminae may be put into resistor B; then the change produced by one lamina Will constitute a finer fraction of the total value of B. Finally, when L is long relative to the thickness b of plate 5, the latter may be curved into a circular cylinder whose base will be formed by edge 52. i

It will be noted here that the laminae 51 of resistor 5, after proper arrangement and packing, are held together by two plastic sheets 56 as shown in exaggerated form in Fig. 4. These sheets may consist simply of gummed paper, or they may be plastic sheets of more or less substantial thickness, which are welded onto the edges of the laminae by first softening (heating) and then setting under pressure. Depending on the total thickness of the sandwiched structure thus produced, on its length in the dimension of X and on the elasticity of the laminae and.

. 11 binding sheets, the structure maybe made flexible enough for bending into a cylindrical surface. Such bending is preferably to be done after the curved edge 53 (y=kX has been cut out and polished, but other orders of procedure may also be applied.

If resistor 5 is thus bent into a cylindrical surface, then rheostat 1 is likewise to be replaced by a solenoid wound around a cylindrical core. Circular variable resistances of this type are quite common in radio practice. Slide 3 is then also replaced by a contactor system which revolves in a circle, and scale 2 is replaced by a circular dial.

Numerous detailed arrangements are possible in such an event. But for the sake of illustration, attention is directed to Fig. 7, wherein semicircles are employed instead of full circles. Rheostat 1 is replaced by the semicircular solenoid 91; resistor plate 5 is bent into semicircle 95; dial 92 takes the place of scale 2. A knob 94 carries a metallic pointer 93 which contacts solenoid 91 through a suitable spur, while its extension guides an armate contact member 933 over semicircular member 95. Pointer 93 is made elastic or is pressed down by springs to insure good contact at 91 and 95. Rollers pressed down by springs (not shown) may further be provided to hold down semicircular strip 933 in good electric contact with member 95.

It will be remembered that the are 95 shown in Fig. 7 corresponds to edge 52 in Fig. 2, and represents the butt ends of the lamina 951. The dimension y runs perpen dicular to the plane of the drawing and in the direction away from the reader, so that knob 94 may be turned freely, without interference from the body of the specialcurve resistor 95. Contact point 34 for the wiring circuit may now be any point on pointer 93, and may run through the axle of knob 96. Contact point is now at 910,

which is the zero-end of solenoid 91, while contact point 55 is, as before, at the end of y which is now in the third dimension away from the reader of Fig. 7.

In Fig. 7, the dial is preferably laid oif not in degrees of arc but in decimal parts of the total active are involved in the resistor B.

It will be understood from all the aforegoing that my invention is susceptible of wide variation as to form, without departing from the spirit thereof. For instance, instead of placing the unit directly under rheostat 1 in Fig. 2, to be joined electrically by a sliding member shaped like a tuning fork, the tworesistances may be placed side-by-side as in Fig. 8 to be operated by a somewhat boat-shaped sliding member 3a. Or they may be placed at an angle with respect to each other, replacing slide 3 by a suitable mechanism achieving the two contacts, in their respective angular directions, simultaneously.

Rollers held down by springs (not shown) may be provided to hold spur 32 and leg 33 of member 3 in firm contact with their respective resistors.

Instead of a scale 2, graduated in decimals to indicate the displacement of slide 3 as a fraction x of the maximum possible displacement, it may be calibrated in any other manner to read for instance millivolts, degrees of arc, pressure in pounds per square inch, or any other quantity of interest in any particular practical problem.

Instead of battery 4 in Fig. 2, a standard cell (of known voltage) may be employed, together with suitably placed, adjustable resistances in series, so as to convert the unit into a direct-reading potentiometer.

In a cascaded system the set of successiveswitches 6 may be replaced by a single bank of switches so designed that it is impossible to throw any particular member of the bank from terminals 61 to terminals 62, without automatically closing the switch in the next unit of the cascade upon terminals 61, so as to insure sealing the entire cascade by fixed resistance R. 7 I

Numerous other changes in detail will be readily apparent to those skilled in the art. A particularly interesting modification of this invention is discussed in the following paragraphs.

Special modification Instead of using for rheostat 1 the traditional straight wire or a uniformly wound solenoid, We may use for this purpose another laminated resistor built on the same general principles as resistor 5, but having as y a straight line, sloping from 0 to y Unlike resistor 5, however, the resistance used here is that of a single element y instead of the integrated sumof the 'ys in a specified region. The details of such a rheostat are shown in Fig. 9, While its use in conjunction with resistor 5 is illustrated in Figs.

In Fig. 9, the resistor-plate 10 is made up of mutually insulated conducting laminae 101, whose butts are lined up and polished to form the straight line 102. This constitutes the :c-axis, which varies from 0 to 1, or the X-axis which varies from .0 to L. (It is clear that x=X/L.) The heights of the laminae are defined by the equation y=m(L-X) or y=mL(l*-x); their ends 01)- posite to the aligned butts therefore outline a straight ine 103 which slopes from y to 0. '(As in the case of resistor 5, line here is used in a relative sense. Actually, the laminae have width b in the dimension perpendicular to the drawing, and edges 102 and 103 are plane surfaces.) W

In welded contact with this sloping edge 103 (by which I mean a fixed and intimate contact) throughout its length is a low-resistance (metallic) conducting strip 104, whose lower end corresponds to point 0 in Fig. 2 and is connected to battery 4. Along the edge 1E2 moves contact-spur 32 at the end of leg 31 of sliding member 3. The current flow follows the arrows shown in Fig. 9. Entering from the battery through point 105, the current flows up the conducting strip 104 to the ordinate (or small group of ordinates) y which is in contact with spur 32, and then out along leg 31. Theoretically, spur 32 should be a knife-edge, so as to contact but one of the ys. Actually, however, it may be a small area, having a finite width along the .r-axis, so that contact is made with a bundle of neighboring laminae. However, the height y,,, of the tallest lamina as well as the thickness and j-value of each must be designed with a particular width of spur 32 in view, and once so designed, the spur must not be replaced by one of different contact area. The calculations involved in the design of this resistor 10 are as follows:

For each lamina of height y, the conductance is wherein j, t, band y have the samemeaning as above, while 1 represents the thickness of each lamina in the direction of the X-axis. The resistance, then, is

If a single lamina is contacted by spur 32, this is the value of A required according to Fig. 1. If more than 1 lamina is involved in the contact, their total conductance is the product of their average multiplied by their number. Their net resistance is therefore approximately equal in value to the resistance of the median y divided by the number of ys contacted. This drop in resistance must be compensated for in the selection of the slopecoeificient m for the expression y==m(L-X).

For instance, let AX=width of spur 32 along the X-axis, and let AX=n't;then' 13 At the higher edge of the plate, y=y and at that point it is required that A=R; hence,

i jbAX Let y=m(L-X); at y,,,, X therefore, y =mL. Inserting in (11),

mL rai RjbAX pL If R, i, b, AX and p have been selected, m can be readily computed. Then finally,

Principles of design In practical design, it is desirable to have the dimensions of the plates and of the same order of magnitude. Letting Y represent the highest ordinate in plate 5, and Y the highest ordinate in plate 10, it is desirable to have both Y and Y roughly equal to L. By the formulas above derived for resistor B, y=kX wherein Rjcb 1fllz L pL and the highest ordinate is at X=L; therefore Y =kL For resistor A, y=m(L-X), wherein RjbAX pL and the highest ordinate is at X :0; therefore, Y =mL. If Y =Y then m=kL. If we want Y and Y both to equal L, we must choose 1 k-Z and m 1 lB B= PB and , Rj b AX=mp L=kp L (l7) Dividing (17) by (16) l A l L 18) (18) then may be considered as the basic formula for design, when resistors A and B are constructed as in Figs. 9 and 10. However, we may still set b =b (which implies equal thickness in plates 10 and 5); then Jul-AX 19) j PB L Assuming that we now choose 25 cm. as a convenient value for L, and 0.25 to 0.5 mm. as a suitable value for AX (which is the width of the contact area of spur 32), then L is about This, then, is the practical value of the left-hand member of (19).

We may still choose j =j (which implies making the ruled conducting lines on the plastic laminae of the same width and thickness for both resistor A and resistor B). Then (19) compels us to choose p and p so as to have a ratio between 0.002 and 0.001. This condition can be satisfied by choosing a graphite ink for resistor B (p=800 microhms) and a silver printing ink for A (p le microhms), or by choosing filament carbon for resistor B (pi4000 microhms) and cadmium for resistor A 8 microhms). Other convenient combinations can be readily selected from tables of resistivity in the literature. (See for instance, Handbook of Chemistry and Physics, 19th edition, pages 1337 to 1343.)

Finer adjustments of the ratio (19) may be made by varying the js as well as the s. For instance, we may print the plastic foil with conducting inks to produce a thin line for the one resistor, while spraying the entire surface of the foil with its respective conducting paint for the other conductor. Or again, we may print the foils in both cases, except make the imprinted lines of different widths or thicknesses in the two cases. For instance, by making the silver line for resistor A about 4/3 times as wide as the graphite line for resistor B keeping the bs equal, as above), we obtain from (19) a ratio of 0.0015; then, if L=24 cm, AX can be made equal to 0.36 mm.

Simplified design procedure The above discussion solves the problem insofar as selecting the requisite materials and mode of printing are concerned. The actual design, however, may still be difiicult, because it would require precise knowledge of the resistivity values of the inks selected and the j-values of the imprinted conducting lines. To avoid this amhiguity, I propose the following procedure for building and shaping resistors A and B.

Having chosen the requisite plastic film (cellulosic film, polyvinyl, methacrylate, etc.), having settled upon the desired values of L, b and AX, having settled upon the printing-ink pair to be chosen (say, graphite and silver), and having estimated approximately the desired width and thickness of the imprinted line, proceed now as follows:

(1) Print the film in two sets with their respective inks, the printed lines being spaced ott at intervals corresponding to the chosen b-value.

(2) Cut the imprinted films transversely of the imprinted lines, to give sheets of width slightly greater than the selected L. (The excess is simply to provide a margin for finer trimming and polishing.)

(3) Chop each sheet (by the aid of a suitable slicer,

chopping machine or power shears) in a direction parallel to the imprinted lines, to give strips of width b.

(4) Stack the strips into piles of convenient height (say, 1 or 2 inches) and press them together (with or without application of heat to soften the plastic) to give compact sticks, which will now have the dimensions of L cm. long, b cm. thick, and perhaps 2 to 4 (or more) cm. wide.

(5) Place several of these sticks together tightly along their width, until their total width equals about L, thus obtaining a square, roughly L cm. on each side and b cm. thick.

(say, methyl methacrylate.) on each face of each square, 7

and apply heat and pressure until the system has become welded, so to speak, into a continuous, single plate.

(7) Trim or polish the edges of each square to perfect it into the dimension L on each side.

(8) Now, taking one of the plates intended for resistor B, apply a pair of negligible-resistance electrodes (i. e. contacting strips of metal, similar to leg 33 above) to the edges of the square which contain the butt ends of the laminae, and using a dry cell, a rnillivoltmeter, standard adjustable resistances or otentiometers (or whatever other apparatus may be needed), measure the resistance of the entire square in the direction of the ruled conducting lines. Let T represent this total resistance.

Then

jsb 1 n T (20) {Proof:

Going back to Equation 1 above, replace y by L and integrate between the limits and L; also, replace 0 by l/ Then (9) Taking one of the laminated plates intended for resistor A, apply to it, on one of the butt-end edges, one of the contacting spurs 32 which have been designed for use with the resistors A of this set in the subsequent assembly of the potentiometer units. (This implies a pre-arranged design of slide-member 3 or at least of its spur 32, and a faithful adherence to this design in the subsequent assembly.) Attach a negligible-resistance electrode to the opposite edge of the plate, and using dry cells, voltmeters, etc., measure the resistance of the plate lengthwise of the ruled lines. (Note that the resistance now is not that of the entire plate, but only of that group of laminae which are contacted. by spur 32.) Let P be the value of the resistance thus measured. Th'en j bAX 1 PAL 13 (21) (Proof: In Equation 1 above, replace y by L and c by l/ Observe that P=l/AC.)

Dividing (21) by (20),

71294 T AX 22 MB P L P Recalling that y=m(L-X), which gives Y ==mL, lay off now, on the square intended for resistor A, an ordinate on the edge intended for X =0; draw a straight line from the ordinate Y to the corner point (X=L, 3 :0). Cut and polish the plate along said straight line, and apply thereto conducting strip 104.

(ll) Compare Equations 14 and 20; then Then instead of pursuing the difiicult task Recalling that for resistor B, y=kX simply plot on the B-plate the parabola RX= p p TL (As a check, observe that when X= L,

Then out out and polish the plate along the plotted curve, and apply thereto conducting strip 54.

It will be clear, of course, that such determination of T, P, k and m will be required only once for each novel design; afterwards, the dimensions may bev duplicated unto replicas of this design by standardized jigs or any suitable means. 7

Furthermore, if especially wanted, both plate and plate 5 may be bent into a semicircular cylinder, scale 2 then being replaced by a dial. But in such event the value of the two Ls will be different for the two resistors, being in fact proportional to the radii of the respective semicircles. The above principles of design must then be modified to take into account the said difference in the values of L and L 25 Control of accuracy Themathematical accuracy of the device of my invention will depend first of all on the number oflaminae in the plates 5 and 10. For instance, i-f L=25 cm., and if the laminae are made up of plastic film whose thickness is 0.01 cm., there will be 2500 such laminae in the range 0 to 1. Inclusion or exclusion of one such lamina under slide 3, determines the smallest change that may be made in the values of resistances A and B laid off. The degree of precision is then 1 in 2500 or 0.04%. To make use of this high degree of precision, it is necessary to provide means for adjusting the position of slide 3 to within 0.01 cm. A screw-feed device would readily achieve this result; such a device is suggested in Figures 10- and 11, wherein the stem of slide 3 is threaded and passes inside a bored block 36 which is held in the fork of support 37. If the bore of this block be threaded, rotation thereof within the confined .space of the fork will advance or recede slide 3 along the xaxis.

However, instead of forming a complete circular body with a threaded bore, block 36 is preferably made up of two circular segments. This is shown in Fig. 12. Segment 360 covers some 300 of arc; or more, and has a 'smooth bore, large enough to rotate freely over stem 30 of slider 3, without engaging its thread. The remainder of the circuit is completed by the wedge-like segment 361, which is threaded on the inner end to mesh with the thread of stem 30. Normally segment 361 is held out of 'mesh with the threaded stem by spring 362, which is fastened to segment 360. This enables free sliding of stem 30 through the bore of block 36.- 'But when a screwfeed motion is desired, wedge 361 is pressed down with the thumb until it engages the thread of stem 30. R0-

. tation of the block, efifected by the thumb and fingers, 0 then imparts a micrometer-type motion to slide 3. Needless to say, segment 360 may be engraved with a scale to indicate the amount of rotation or the amount of forward motion imparted to slide 3.

Other variations in detail, will be readily apparent to those skilled in the art. For instance, scale 2 may be placed between the two laminated "plates, or in any other convenient position. Rollers, held down by springs, may be provided topress leg 33 and spur 32 tightly against the butts of their respectively contacted laminae. Proper supports or housing may be provided for -the various elements of the device, and the latter may be insulated from such supports where necessary or, on the contrary, may be grounded through them.

Instead of laying out the lamina in plates 5 and 10 strictly in straight line formation, they may be bent into arcs or sinuous curves, provided their actual lengths, not their height above the x-axis, is taken into account in laying oil the curves y=kX and y=m(L-X). This principle is illustrated in Fig. 13, wherein the dotted outline shows the modified shape of plate before cutting out the curve y, while the solid outline shows the finished resistor B. The ends 512 of laminae 51 are bent backward slightly to expose the printed face of each lamina for slightly better contact with leg 33 of slide 3. The same principle may be applied, if desired, to the ends 513 of the lamina, or to the laminae of resistor A.

Furthermore, the imprinted conducting lines may be flared out in the process of printing to form a 't at that end of each lamina which is intended to make contact with slide 3, so as to provide a wider line of contact with leg 33 or spur 31. However, the height of the widened butt thus formed should be discounted (because of its lower resistance value) in computing the curve y=kX or y=m(LX).

instead of finishing off edge 52 of plate 5 in the form of a plane surface, of length L and width b, this edge be made trough shaped or slightly grooved, with the butt ends of the imprinted conducting lines at the bottom of the trough or groove, so as to protect these ends against abrasion in handling. The contacting member 33 will then naturally be given a convex or V-shaped edge on the lower side, so as to insure proper contact with said conducting butts. Similar grooving may also be applied to any of the other edges 53, 102 and 103 above discussed, with corresponding convexing compensation in the contactors 32, 54 and 104.

In making the measurements as above of a sample plate from each set to determine the values of ink and

j bAX PA other shapes than a square may be measured (for instance, a right isosceles triangle obtained by cutting one such square along its diagonal), provided corresponding revision is made in the mathematical relations involved.

Surface printed modification Still another modification according to this invention :uses for variable resistances A and B planoprinted resistors. These resistances differ from those already described in that they are not laminated. Instead, a series of parallel lines are printed, in close proximity, with a conducting ink on a non-conducting surface. Such a resistor, suitable for variable resistance 8 above described, is shown in Figs. 14 and 15, the latter being a section of .the former along line 15-i5.

A plate 12, which corresponds to plate 5 in Fig. 3, is imprinted with a series of parallel, vertical lines 121 by means of a conducting ink. The plate 12 may be of .ceramic material, and the conductingink may be a silver ink, in which case the plate is first imprinted and then .fired, to produce an intimately bonded imprint. Or the conducting lines may be imprinted on a sheet of plastic material, and the latter may then be bonded, for instance by adhesive, to a plate 12 may of any conventent material such as cardboard, wood, plastic or metal. in the latter event, the imprinted sheet may be initially of rectangular shape, and may be subsequently cut out into the shape indicated in Fig. 14 as required for its intended use.

The lines and their spacing in Fig. 14 are indicated diagrammatically. Actually, the lines are much finer, and their spacing much closer than shown in this figure. In the printing of plastic sheets with copper foil by the etching method, it is not uncommon to make the lines 0.00135 inch thick and 0.005 inch wide. (Modern Plastics, August 195.], pages 99 to ill.) in. printing paper 18 with carbon inks, very fine printing may likewise be achieved.

Let b represent the period of the pattern, that is, the distance from the leading edge of one line to the leading edge of the next line. Then each element of height y and width b on the imprinted sheet is equivalent to one lamina in the laminated modifications. The resistance of such element is EL AA where AA represents the cross section of one of the imprinted lines. For instance, if lines of copper foil of the above width and thickness are imprinted,

i. e. 6.75 l0' sq. in. or about 43.5 10- -sq. cm. The conductor being copper, p is about 1.7 1O The resistance of each element, then, is about 0.039 If y is given a height of 25 cm. at its highest point, the resistance of the highest line is about 1 ohm. This design then is suitable for making resistor A. e

For resistor B, a carbon or graphite ink is to be chosen, and its -value will be of the order of 3000 to 4000 Comparing this with Formula 1, except restating the latter as AC jbAX Pl! we observe that we need merely to replace the expression J' y to convert (1a) into (24). The same replacement if carried consistently throughout the remaining Formulas 2 to 23 will yield corresponding formulas for design in the instant case, that is, using surface-imprinted plates as resistors.

With a printed plate of the above type, contact may be made with a portion of the imprinted line instead of with its butt end. For this purpose, each y on plate 12 is laid off with small extensions Z1 and Z2 on the ends. Slide 133 along edge 122 (corresponding to leg 33 of slide 3 in Fig. 3), then is given a cross-sectional shape as shown in Fig. 15, whereby it makes contact on the surface of plate 12 along the extensions 2 Along edge 123, likewise, a fixed contact member 124 is provided which bends over the surface of the plate and covers up the extensions Z2 of the ordinates. Obviously, the mathematical relation y =kX above postulated, contemplates as y the length of each ordinate between said extensions (exclusive of the latter).

The imprinted y-lines need not be perpendicular to the x-axis. Instead, they may be inclined, as shown in Fig. 16, provided it is remembered that it is the length of each y (exclusive of its extensions z), and not its vertical projection, that should correspond to the basic formulas above derived. Of course, where the conducting lines are imprinted on a thin, plastic sheet or paper, it is not necessary to print the lines in inclined form. Instead, the lines may be printed perpendicular to the edge of the sheet, but the latter may then be cut out properly to give the shapes indicated in Fig. 16.

Plate 14 in Fig. 16 corresponds to laminated plate 10 in Fig. 9, except that here we use a surface imprinted plate in lieu of a laminated composite plate. The extensions Z Z here, and the fixed contact member 144 serve here the same purpose as the respective elements Z1, 2 and 124 in Fig. 14. Sliding contact here is made with spur 132 of slide 13, which is shown in greater detail in Fig. 18. Contact points 125 and 145 play here the same role as points 55 and 105 in Figs. 3 and 9.

The advantage of using inclined lines as in Fig. 16 is that it condenses the outline of the combination into a rectangular, almost square, space. This fact, together with the fiat, and relatively thin, aspect of the resistor plates makes the modification of Fig. 16 particularly advantageous for cascading several units together; for then the several units may be stacked on top of each other, producing a compact cascaded assembly.

I claim as my invention:

1. In a device of the character described, the combination of a variable resistance A with a compound resistance consisting of a variable resistance B in parallel with a fixed resistance R, said resistance A being connected in series with said compound resistance, and means associated with said combination for varying resistances A and B in unison whereby to maintain between them the mathematical relation AB=R x, x being an arbitrary number of value between zero and l, and whereby furthermore the total resistance of the combination is maintained at the fixed value R, regardless of the value of x selected.

2. In a device of the character described, a circuit comprising a source of E. M. F. in series with a system of resistances, said system including a variable resistance A, a variable resistance B and a fixed resistance R, said resistance B and fixed resistance R being connected in parallel to form the subcombination BR, and said variable resistance A being connected in series with said subcombination 1i, and means for varying simultaneously the values of resistances A and B, said means being designed to move in a path of total length L and being adapted to halt at any position xL along'said path, x being a number of value between zero and 1, said variable resistance A being designed to assume the value R(lx) when said moving means is at position xL, and said variable resistance B being designed to assume the value ance of the series-parallel system A-lil is maintained steady at the value R regardless of the value of x, and that the potential difierence across the terminals of said subcombination BR is equal to xE, E being the potential difference between the terminals of said system A-BR.

3. A device of the potentiometer type comprising in combination a variable resistance A in series with a branched resistance a, the latter consisting of a variable resistance B in parallel with a fixed resistance R; means to impress a voltage E upon the terminals of said series system A-B R; a sliding contact member adapted to contact simultaneously resistancesA and B whereby to determine their respective etfective resistance values by the position of said contact member, and conducting elements connected to the terminals of said resistance R whereby the potential difference e across said resistance R may be measured or utilized.

4. A combination as in claim 3, said resistances A and B being so designed that the total resistance of the combination AB R is maintained at a steady value equal to R regardless of the position of said sliding contact member.

5. A combination as in claim 3 comprising further a switch associated with said fixed resistance R, whereby the latter may be excluded from the system and replaced by a circuit having a net resistance equal to R.

6. A combination as in claim 3 comprising further another potentiometer type device of the same structure, and a switch associated therewith and with resistance R of the first potentiometer circuit, whereby said resistance R may be excluded from the first potentiometer circuit and replaced by said second potentiometer circuit.

7. A combination as in claim 6 including further means whereby said fixed resistance R is shifted from the circuit of the first potentiometer into the circuit of the second, while the terminals of the system A-BR in said second potentiometer circuit are switched into the first circuit in the position formerly occupied by said fixed resistance R.

8. A cascaded system of potentiometer circuits comprising a plurality of electrically similar potentiometer units, each unit being composed of a variable resistance A in series with a variable resistance B, each next unit in the system having the terminals of its series combination AB shunted across the terminals of resistance B in the immediately preceding unit; a fixed resistance R shunted across the terminals of the last resistance B of the system, and means associated with the variable resistances A and B of each unit whereby to maintain the elfective reistance of the A-B series in each unit at a fixed value R as long as each intermediate B resistance has shunted across its terminals the respectively following unit above set forth and as long as the B resistance of the last unit has shunted across its terminals said fixed resistance R or an electrical system of equal resistance.

9. A cascaded system as in claim 8, said last unit having switching means associated with said fixed resistance R whereby to replace the latter by an electrical circuit of optional composition.

References Cited in the file of this patent UNITED STATES PATENTS 1,940,102 Robertson Dec. 19, 1933 2,453,462 Sellers Nov. 9, 1948 2,625,633 Warsher Ian. 13, 1953 2,684,463 Wilentchik July 20, 1954 FOREIGN PATENTS- 449,887 Germany Sept. 8, 1927 UNITED STATES PATENT OFFICE Certificate of Correction Patent N 0. 2,831,158 April 15, 1958 David Katz It is hereby certified that error appears in the printed specification of the above numbered patent requiring'correction and that the said Letters Patent should read as corrected below.

Column 3, line 65, for

read 1 column 7, line 10, after meter insert a comma; column 13, lines 13 and 48,

for m, in each occurrence, read -m==-; line 55, for P read column 17, line 64, for may read -made; same line, for conventent read conven1ent.

Signed and sealed this 5th day of August 1958.

Attest: KARL H. AXLINE, ROBERT C. WATSON, Attestz'ng Oyficer. Commissioner of Patents.

UNITED STATES PATENT OFFICE Certificate of Correction Patent No. 2,831,158 April 15,1958 David Katz It is hereby certified that error appears in the printed specification of the above numbered patent requiring correction and that the said Letters Patent should read as corrected below.

Column 3, line 65, for

5 read & R R

column 7 line 10, after meter insert a comma; column 13, lines 13 and 48,

for m in each occurrence read m=-' line 55 for P read a a a a P 7 column 17, line 64, for may read made-; same line, for conventent read conven1ent.

Signed and sealed this 5th day of August 1958.

Attest: v KARL H. AXLINE, ROBERT C. WATSON, Attesting Ofioer. Gammissioner of Patents. 

